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Numerical Recovering of a Speed of Sound by the BC-Method in 3D
Abstract
We develop the numerical algorithm for solving the inverse problem for
the wave equation by the Boundary Control method. The problem, which we
refer to as a forward one, is an initial boundary value problem for the
wave equation with zero initial data in the bounded domain. The inverse
problem is to find the speed of sound c(x) by the measurements of waves
induced by a set of boundary sources. The time of observation is assumed
to be greater then two acoustical radius of the domain. The numerical
algorithm for sound reconstruction is based on two steps. The first one
is to find a (sufficiently large) number of controls {f_j} (the basic
control is defined by the position of the source and some time delay),
which generates the same number of known harmonic functions, i.e.
Δ {u_j}(.,T) = 0 , where {u_j} is the wave generated by the
control {f_j} . After that the linear integral equation w.r.t. the speed
of sound is obtained. The piecewise constant model of the speed is used.
The result of numerical testing of 3-dimensional model is presented.
An inverse coefficient problem for time-dependent wave equation in three dimensions is under consideration. We are looking for a spatially varying coefficient of this equation knowing special time integrals of the wave field in an observation domain. The inverse problem has applications to the reconstruction of the refractive index of an inhomogeneous medium, as well as to acoustic sounding, medical imaging, etc. In the article, a new linear three-dimensional Fredholm integral equation of the first kind is introduced from which it is possible to find the unknown coefficient. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require significant computational resources and a long solution time. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the wave field. Typical results of solving this three-dimensional inverse problem on a personal computer for simulated data demonstrate high capabilities of the proposed algorithm.
We present the results on numerical testing of the Boundary Control Method in the sound speed determination for the acoustic equation on semiplane. This method for solving multidimensional inverse problems requires no a priory information about the parameters under reconstruction. The application to the realistic Marmousi model demonstrates that the boundary control method is workable in the case of complicated and irregular field of acoustic rays. By the use of the chosen boundary controls, an `averaged' profile of the sound speed is recovered (the relative error is about ). Such a profile can be further utilized as a starting approximation for high resolution iterative reconstruction methods.
In this paper we develop а numerical algorithm for solving inverse dy-namical problem: recovering an absorption by the Boundary Control method. The absorption coefficient is determined independently of a speed of sound, which can be an arbitrary smooth positive function. The results of numerical experiments are represented.
We introduce a new algorithm to construct travel time distances between a
point in the interior of a Riemannian manifold and points on the boundary of
the manifold, and describe a numerical implementation of the algorithm. It is
known that the travel time distances for all interior points determine the
Riemannian manifold in a stable manner. We do not assume that there are sources
or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map,
or its restriction, as our data. Our algorithm is a variant of the Boundary
Control method, and to our knowledge, this is the first numerical
implementation of the method in a geometric setting.
The boundary control method (BC-method) is an approach to inverse problems based on their relations to control theory. Originally, it was proposed for solving multidimensional
problems (see [2]) and therefore such an approach cannot be simple: the BC-method uses asymptotic methods in PDEs, functional analysis, control
and system theory, etc. The aim of this lecture course is to provide possibly elementary and transparent introduction to the
BC-method. Namely, we present its 1-dimensional version on example of two dynamical inverse problems.
In this paper we develop numerical algorithm for solving inverse problem for the wave equation using Boundary Control method. The results of numerical experiments are represented.
We consider an inverse problem for the wave equation: find the speed of sound given a reaction operator on a part of the boundary. An approach for solving the problem is proposed based on the boundary control method developed by M. I. Belishev. In this case the method applies by a choice of retarded sources of a special form. The reconstruction of the speed sound is carried out in a Riemann hemisphere and reduces to an equation of the Gel’fand-Levitan type for the unknown boundary source, and inversion of the Fourier transform.
The review covers the period 1997–2007 of development of the boundary control method, which is an approach to inverse problems based on their relations to control theory (Belishev 1986). The method solves the problems on unknown manifolds: given inverse data of a dynamical system associated with a manifold it recovers the manifold, the operator governing the system and the states of the system defined on the manifold. The main subject of the review is the extension of the boundary control method to the inverse problems of electrodynamics, elasticity theory, impedance tomography, problems on graphs as well as some new relations of the method to functional analysis and topology.